The value of `(i+sqrt3)^(100)+(i-sqrt3)^(100)+2^(100)` is

Options

A. 1
B. `-1`
C. 0
D. 2

Answer

To find the value of \( (i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} + 2^{100} \), we can follow these steps:

### Step 1: Express in terms of roots of unity
We can express \( i + \sqrt{3} \) and \( i - \sqrt{3} \) in a polar form. Notice that:

\[ i + \sqrt{3} = \sqrt{3} + i = 2 \left( \frac{\sqrt{3}}{2} + \frac{1}{2} i \right) \]
This can be represented as \( 2 \text{cis} \left( \frac{\pi}{6} \right) \) since \( \frac{\sqrt{3}}{2} = \cos \frac{\pi}{6} \) and \( \frac{1}{2} = \sin \frac{\pi}{6} \).

Similarly, for \( i - \sqrt{3} \):

\[ i - \sqrt{3} = -\sqrt{3} + i = 2 \left( -\frac{\sqrt{3}}{2} + \frac{1}{2} i \right) \]
This can be represented as \( 2 \text{cis} \left( \frac{5\pi}{6} \right) \).

### Step 2: Raise to the power of 100
Now we can raise both expressions to the power of 100:

\[ (i + \sqrt{3})^{100} = \left( 2 \text{cis} \left( \frac{\pi}{6} \right) \right)^{100} = 2^{100} \text{cis} \left( \frac{100\pi}{6} \right) = 2^{100} \text{cis} \left( \frac{50\pi}{3} \right) \]

\[ (i - \sqrt{3})^{100} = \left( 2 \text{cis} \left( \frac{5\pi}{6} \right) \right)^{100} = 2^{100} \text{cis} \left( \frac{500\pi}{6} \right) = 2^{100} \text{cis} \left( \frac{250\pi}{3} \right) \]

### Step 3: Simplify the angles
Now we simplify the angles:

\[ \frac{50\pi}{3} = 16\pi + \frac{2\pi}{3} \quad \text{(since } 16\pi \text{ is a multiple of } 2\pi\text{)} \]
Thus, \( \text{cis} \left( \frac{50\pi}{3} \right) = \text{cis} \left( \frac{2\pi}{3} \right) \).

Similarly,

\[ \frac{250\pi}{3} = 83\pi + \frac{1\pi}{3} \quad \text{(since } 83\pi \text{ is a multiple of } 2\pi\text{)} \]
Thus, \( \text{cis} \left( \frac{250\pi}{3} \right) = \text{cis} \left( \frac{\pi}{3} \right) \).

### Step 4: Combine the results
Now we can combine the results:

\[ (i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} = 2^{100} \left( \text{cis} \left( \frac{2\pi}{3} \right) + \text{cis} \left( \frac{\pi}{3} \right) \right) \]

Using the property of cis:

\[ \text{cis} \left( \frac{2\pi}{3} \right) + \text{cis} \left( \frac{\pi}{3} \right) = 2 \cos \left( \frac{2\pi}{3} \right) = 2 \left( -\frac{1}{2} \right) = -1 \]

### Step 5: Final expression
Thus, we have:

\[ (i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} = 2^{100} (-1) = -2^{100} \]

Now we add \( 2^{100} \):

\[ -2^{100} + 2^{100} = 0 \]

### Final Answer
The value of \( (i + \sqrt{3})^{100} + (i - \sqrt{3})^{100} + 2^{100} \) is \( \boxed{0} \).